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      <h1 class="mume-header" id="sha3-%E5%9F%BA%E4%BA%8E%E7%BD%AE%E6%8D%A2%E7%9A%84%E5%93%88%E5%B8%8C%E5%92%8C%E5%8F%AF%E6%89%A9%E5%B1%95%E8%BE%93%E5%87%BA%E5%87%BD%E6%95%B0">SHA3 &#x57FA;&#x4E8E;&#x7F6E;&#x6362;&#x7684;&#x54C8;&#x5E0C;&#x548C;&#x53EF;&#x6269;&#x5C55;&#x8F93;&#x51FA;&#x51FD;&#x6570;</h1>

<p><span id="toc"></span></p>
<ul>
<li><a href="#sha3-%E5%9F%BA%E4%BA%8E%E7%BD%AE%E6%8D%A2%E7%9A%84%E5%93%88%E5%B8%8C%E5%92%8C%E5%8F%AF%E6%89%A9%E5%B1%95%E8%BE%93%E5%87%BA%E5%87%BD%E6%95%B0">SHA3 &#x57FA;&#x4E8E;&#x7F6E;&#x6362;&#x7684;&#x54C8;&#x5E0C;&#x548C;&#x53EF;&#x6269;&#x5C55;&#x8F93;&#x51FA;&#x51FD;&#x6570;</a>
<ul>
<li><a href="#hashtoc">Hash</a></li>
<li><a href="#xoftoc">XOF</a></li>
<li><a href="#%E7%AC%A6%E5%8F%B7%E8%AF%B4%E6%98%8Etoc">&#x7B26;&#x53F7;&#x8BF4;&#x660E;</a></li>
<li><a href="#%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99toc">&#x53C2;&#x8003;&#x8D44;&#x6599;</a></li>
</ul>
</li>
</ul>
<h2 class="mume-header" id="hashtoc"><a href="#toc">Hash</a></h2>

<p>&#x8BB0;&#x6709;&#x6D88;&#x606F;<span class="mathjax-exps">$M$</span>, SHA-3&#x5B9A;&#x4E49;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; SHA3-224(M) = Keccak[448](M || 01, 224) \\ &amp; SHA3-256(M) = Keccak[512](M || 01, 256) \\ &amp; SHA3-384(M) = Keccak[768](M || 01, 384) \\ &amp; SHA3-512(M) = Keccak[1024](M || 01, 512) \\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; Keccak[c] = Sponge[Keccak_p[1600, 24], pad, 1600-c] \\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} pad(x,m) = 1 || 0^j || 1, \quad j = (-m-2) \mod x,\quad x \in Z^{+}, m \in N \\ \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} Sponge[k[b, n_r], pad, r](N, d): \\ &amp; P = N || pad(r, len(N)) \\ &amp; n = len(P) / r \\ &amp; c = b-r \\ &amp; P_i = Trunc_{i*r\dots (i+1)*r}(P) \\ &amp; S = 0^b \\ &amp; for \ i \ in \ 0..n \\ &amp; \quad S = k(S \oplus (P_i || 0^c)) \\ &amp; end \\ &amp; Z = &quot;&quot;\\ &amp; loop \\ &amp; \quad Z = Z || Trunc_{0\dots r}(S) \\ &amp; \quad if \quad d \le len(Z) \\ &amp; \quad \quad return\quad Trunc_{0\dots d}(Z)\\ &amp; \quad else \\ &amp; \quad \quad S = k(S) \\ &amp; \quad end \\ &amp; end \\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} Keccak[b, n_r](S),\ assert(len(s)) = b: \\ &amp; M_{S\rightarrow A}: A[x,y,z] = S[w*(5y+x)+z], x\in [0,5), y\in [0, 5), z\in[0, b) \\ &amp; Rnd(A, i_r) = \iota(\chi(\pi(\rho(\theta(A)))), i_r)\\ &amp; for\ i_r\ in (12+2l-n_r)..(12+2l) \\ &amp; \quad A = Rnd(A, i_r) \\ &amp; end \\ &amp; return\quad M^{-1}_{S\rightarrow A} \\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} \theta(A): \\ &amp; C[x,z] = A[x,0,z]\oplus A[x,1,z]\oplus A[x,2,z]\oplus A[x,3,z]\oplus A[x,4,z] \\ &amp; D[x,z] = C[(x-1)\mod 5, z] \oplus C[(x+1)\mod 5, (z-1)\mod w] \\ &amp; return\quad A[x,y,z]\oplus D[x,z]\\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} \rho(A): \\ &amp; A&apos;[0,0,z] = A[0,0,z], \forall z\in [0,w) \\ &amp; (x,y) = (1,0) \\ &amp; for\ t\ in\ [0, 24)\\ &amp; \quad A&apos;[x,y,z] = A[x,y,(z-(t+1)(t+2)/2)\mod w], \forall z\in [0,w) \\ &amp; \quad (x,y) = (y,(2x+3y)\mod 5) \\ &amp; end \\ &amp; return\quad A&apos;\\ &amp;\quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} \pi(A): \\ &amp; A&apos;[x,y,z] = A[(x+3y)\mod 5, x, z], \forall x\in[0,5), \forall y\in[0,5), \forall z\in [0,w) \\ &amp; return\quad A&apos; \\ &amp;\quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} \chi(A): \\ &amp; A&apos;[x,y,z] = A[x,y,z]\oplus ((A[(x+1)\mod 5, y, z] \oplus 1) \&amp; A[(x+2)\mod 5, y, z]), \forall x\in[0,5), \forall y\in[0,5), \forall z\in [0,w) \\ &amp; return\quad A&apos; \\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} \iota(A,i_r): \\ &amp; A&apos;[x,y,z] = A[x,y,z], \forall x\in[0,5), \forall y\in[0,5), \forall z\in [0,w) \\ &amp; RC = 0^w \\ &amp; for\ j\ in\ [0,l) \\ &amp; \quad RC_{j\dots(j+1)} = rc(j+7i_r)\\ &amp; end \\ &amp; A&apos;[0,0,z] = A&apos;[0,0,z]\oplus RC_{z\dots(z+1)}, \forall z\in [0,w)\\ &amp; return\quad A&apos; \\ &amp; \quad \end{aligned}$$</div><p></p>
<p></p><div class="mathjax-exps">$$\begin{aligned} rc(t): \\ &amp; tmp = t\mod 255 \\ &amp; if\ tmp = 0 \\ &amp; \quad return\quad 1 \\ &amp; end \\ &amp; R = 10000000 \\ &amp; for\ i\ in\ 1..(tmp+1) \\ &amp; \quad R = 0 || R \\ &amp; \quad R_{0\dots 1} = R_{8\dots 9}\\ &amp; \quad R_{4\dots 5} = R_{8\dots 9}\\ &amp; \quad R_{5\dots 6} = R_{8\dots 9}\\ &amp; \quad R_{6\dots 7} = R_{8\dots 9}\\ &amp; \quad R = Trunc_{0\dots 8}(R)\\ &amp; end \\ &amp; return \quad Trunc_{0\dots 1} \\ &amp; \quad \end{aligned}$$</div><p></p>
<table>
<thead>
<tr>
<th style="text-align:center">b</th>
<th style="text-align:center">w</th>
<th style="text-align:center">l</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:center">25</td>
<td style="text-align:center">1</td>
<td style="text-align:center">0</td>
</tr>
<tr>
<td style="text-align:center">50</td>
<td style="text-align:center">2</td>
<td style="text-align:center">1</td>
</tr>
<tr>
<td style="text-align:center">100</td>
<td style="text-align:center">4</td>
<td style="text-align:center">2</td>
</tr>
<tr>
<td style="text-align:center">200</td>
<td style="text-align:center">8</td>
<td style="text-align:center">3</td>
</tr>
<tr>
<td style="text-align:center">400</td>
<td style="text-align:center">16</td>
<td style="text-align:center">4</td>
</tr>
<tr>
<td style="text-align:center">800</td>
<td style="text-align:center">32</td>
<td style="text-align:center">5</td>
</tr>
<tr>
<td style="text-align:center">1600</td>
<td style="text-align:center">64</td>
<td style="text-align:center">6</td>
</tr>
</tbody>
</table>
<h2 class="mume-header" id="xoftoc"><a href="#toc">XOF</a></h2>

<p>&#x8BB0;&#x6709;&#x6D88;&#x606F;<span class="mathjax-exps">$M$</span>, &#x8981;&#x6C42;XOF&#x8F93;&#x51FA;&#x7684;&#x4E8C;&#x8FDB;&#x5236;&#x4E32;&#x4F4D;&#x957F;&#x5EA6;&#x4E3A;<span class="mathjax-exps">$d$</span>, &#x5219;SHAKE&#x5B9A;&#x4E49;&#x5982;&#x4E0B;:</p>
<p></p><div class="mathjax-exps">$$\begin{aligned} &amp; SHAKE128(M, d) = Keccak[256](M||1111, d) \\ &amp; SHAKE256(M, d) = Keccak[512](M||1111, d) \\ \end{aligned}$$</div><p></p>
<h2 class="mume-header" id="%E7%AC%A6%E5%8F%B7%E8%AF%B4%E6%98%8Etoc"><a href="#toc">&#x7B26;&#x53F7;&#x8BF4;&#x660E;</a></h2>

<ul>
<li><span class="mathjax-exps">$x||y$</span>: <span class="mathjax-exps">$y$</span>&#x62FC;&#x63A5;&#x5728;<span class="mathjax-exps">$x$</span>&#x4E4B;&#x540E;;</li>
<li><span class="mathjax-exps">$0^j$</span>: &#x8868;&#x793A;<span class="mathjax-exps">$j$</span>&#x4E2A;&#x4E8C;&#x8FDB;&#x5236;&#x4F4D;<code>0b0</code>&#x62FC;&#x63A5;&#x5728;&#x4E00;&#x8D77;;</li>
<li><span class="mathjax-exps">$Trunc_{i\dots j}(X)$</span>: &#x8868;&#x793A;&#x622A;&#x53D6;<span class="mathjax-exps">$X$</span>&#x7684;<span class="mathjax-exps">$[i, j)$</span>&#x4E4B;&#x95F4;&#x7684;<span class="mathjax-exps">$j-i$</span>&#x4F4D;. &#x5176;&#x4E2D;, <span class="mathjax-exps">$j\gt i \ge 0, j \le len(X)$</span>;</li>
<li><span class="mathjax-exps">$X_{i..j}$</span>: &#x8868;&#x793A;<span class="mathjax-exps">$X$</span>&#x7684;&#x7B2C;<span class="mathjax-exps">$i$</span>&#x4F4D;&#x5230;&#x7B2C;<span class="mathjax-exps">$j$</span>&#x4F4D;&#x4E4B;&#x95F4;&#x7684;<span class="mathjax-exps">$j-i$</span>&#x4E2A;&#x4E8C;&#x8FDB;&#x5236;&#x4F4D;;</li>
<li><span class="mathjax-exps">$X\oplus Y$</span>: <span class="mathjax-exps">$X$</span>&#x6309;&#x4F4D;&#x5F02;&#x6216;<span class="mathjax-exps">$Y$</span>;</li>
<li><span class="mathjax-exps">$M_{S\rightarrow A}$</span>: &#x8868;&#x793A;&#x4E00;&#x4E2A;&#x6620;&#x5C04;, &#x5C06;&#x96C6;<span class="mathjax-exps">$S$</span>&#x6620;&#x5C04;&#x5230;<span class="mathjax-exps">$A$</span>. <span class="mathjax-exps">$M^{-1}_{S \rightarrow A}$</span>&#x8868;&#x793A;&#x5176;&#x9006;&#x6620;&#x5C04;;</li>
</ul>
<h2 class="mume-header" id="%E5%8F%82%E8%80%83%E8%B5%84%E6%96%99toc"><a href="#toc">&#x53C2;&#x8003;&#x8D44;&#x6599;</a></h2>

<ul>
<li>FIPS 202(SHA-3 Standard: Permutation-Base Hash and Extendable-Output Functions);</li>
<li><a href="https://www.cnblogs.com/mengsuenyan/p/12697811.html">SHA&#x5B89;&#x5168;&#x6563;&#x5217;&#x7B97;&#x6CD5;</a>;</li>
</ul>

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